# What do variables with a numerical subscript mean in astronomy?

I am wondering if anyone can help me understand what a variable with a subscript on it means. The paper: B.F. Schutz, Nature 323, 310 (1986). The variable in question is this distance $r_{100}$ and the $f_{100}$, which I assume is related. I'm sure this is some standard notation, but I am having trouble finding a source as google returns information about atomic numbers and such. The paper is related to cosmology/relativity if this is a field related notation.

Consider a binary at a distance $100 r_{100}\:\mathrm{Mpc}$, with total mass $m_\mathrm{T}M_{\odot}$ and reduced mass mass $\mu M_{\odot}$, emitting waves at a frequency $100f_{100}\:\mathrm{Hz}$

Later in the paper there is another notation similar which is defined has $h_{23} = h \times 10^{23}$, which makes sense as this $h$ is typically of $O(10^{-23})$. However, I don't believe that this would be the same notation for $r$ and $f$ as $r_{100} = r\times 10^{100}$ seems absurd, as the universe is only of $O(10^{27})$ centimeters (already a silly unit to measure distances of these scales, 63 orders smaller is 63 orders sillier).

$$\langle h_{23}\rangle=\langle h\rangle\times 10^{23}$$

Possible solution: I have come to the conclusion that perhaps the more logical interpertation of this notation is as follows: $r_{100} = \frac{r}{100 \textrm{ Mpc}}$ and $f_{100} = \frac{f}{100 \textrm{ Hz}}$.

This notation is mostly used in astronomy (or at least I haven't seen it elsewhere), and it is used to de-dimensionalize variables, or otherwise get rid of inconvenient units and exponents, whilst keeping track of what standard each variable is measured in. This is not really a unified notation; instead, it is a bunch of different variables the author have defined, which have a stylistic connection but no uniform interpretation.

Such notation is usually fluid, and the two uses you show - $h_{23}$ and $r_{100}$ - are different, with the first subscript denoting the exponent (i.e. $h_{23}=10^{23}h$) and the second one denoting the yardstick (i.e. $r_{100}=r/100\:\mathrm{Mpc}$). As such, there is in general no way to tell what a given notation of the form $x_n$ means in a given paper, and each document that uses this notation must - and does - include an appropriate definition of every such symbol.

For a similar example see this thread, where the paper has de-dimensionalized the Hubble constant as $h_{50}=H_0/(50\: \mathrm{km\: s^{-1}Mpc^{-1}})$, giving a dimensionless parameter which is used to scale other units (where measurements of distance and related quantities are inferred from redshifts, so changes in the value of $H_0$ would change the reported values), while also making it obvious that the scaling parameter is essentially Hubble's constant, with the unit yardstick in the subscript.

• For context, this does show up in other fields also. In condensed phase explosives they do a test called a drop test, where they drop a set weight from progressively higher heights onto an explosive sample. The height at which 50% of the samples explode is the result reported, designated as $h_{50}$. In turbulence, we also sometimes denote similarity parameters with subscripts to indicate something. Like $\delta_{90}$ is the location where a mixing layer is 90% of it's maximum thickness. Etc.. It's all very case specific as you mention. Jan 19, 2016 at 22:27
• @tpg2114 thankfully I've never really been in the explosives testing business, but that's interesting to note. Jan 19, 2016 at 22:29